where \(\beta =-(3+\alpha )\)  and \(\alpha \)  is a non-zero parameter. For \(\alpha =-3\) , the model is reduced to the standard slow-roll. The need to go beyond of the standard slow-roll approximation goes back to the ultra slow-roll regime where the term of \(\ddot{\varphi }\)  is non-negligible in the Klein-Gordon equation as \(\ddot{\varphi }=3H\dot{\varphi }\) . This class of inflationary models shows a finite value for the non-decaying mode of curvature perturbations and can be expressed with the familiar form of standard inflation [1]}. Although the ultra slow-roll model predicts a large \(\eta \)  because of the non-negligible term \(\ddot{\varphi }\) , it is not able to solve the \(\eta \)  problem introduced in supergravity for the hybrid inflationary models [2]}. Moreover, the inflationary solutions of the ultra model are situated in the non-attractor phase of inflation but show a scale-invariance for the scalar perturbation spectrum. Besides, the main problem of ultra models is that the non-Gaussinaity consistency relation of single field models is violated in the presence of ultra condition through super-Hubble evolution of the scalar perturbation [3]}. Despite the above problems, the ultra slow-roll inflationary model is really interesting since it reveals the plenty of unknown and unexplained issues around the inflationary dynamics. Fast-roll models are known as another class of inflationary models introduced in the beyond of the slow-roll approximation in which a fast-rolling stage is considered at the start of inflation and will be connected to the standard slow-roll only after a few e-folds [4]}, [5]}, [6]}. This approach can be considered as a correction to slow-roll models in order to improve the consistency with observations.
